SBIR and STTR Extension Act of 2022 This bill reauthorizes through FY2025 and modifies the Small Business Innovation Research (SBIR) program, the Small Business Technology Transfer (STTR) program, and related pilot programs. The SBIR and STTR programs are administered by various federal agencies and provide competitive awards for domestic small businesses to conduct research and development projects that have the potential for commercialization. The bill requires agencies with an SBIR or STTR program to assess the security risks presented by applicants with financial ties or obligations to certain foreign countries. The programs may not make awards to businesses with certain connections to foreign entities. Additionally, businesses with more than a specified number of prior awards must meet expanded minimum performance standards. The Small Business Administration's Office of Inspector General must audit the small businesses subject to these expanded performance standards. The bill also establishes various reporting requirements, including directing the Department of Defense, the Department of Energy, the Department of Health and Human Services, and the National Science Foundation to assess the national security and research and integrity risks of the SBIR and STTR programs. The Government Accountability Office must evaluate and report on aspects of the programs such as the subcontracting practices of award recipients.